Integral graphs and (k,τ)-regular sets

A subset of the vertex set of a graph G, W⊆V(G), is a (k,τ)-regular set if it induces a k-regular subgraph of G and every vertex not in the subset has τ neighbors in it. In this paper we deal with the existence of (k,τ)-regular sets associated with all distinct eigenvalues. We show some families that have this property and we give some results concerning the existence of such sets considering restrictions on the symbol of circulant graphs.